You are here

discontinuities in mesh free methods

Primary tabs

I wish to ask where to find literature about introducing discontinuities in the shape functions to simulate cracks in mesh free methods.

I found the visibility criterion, the diffraction method and the transparency method referred in the (I think vey good) survey by Fries and Matthies "classification and overview of meshdree methods" but nothing else.

For modeling discrete cracks, partition of unity based methods might be preferable (rather than meshfree). You can see this post and the references therein for further details on this subject. A quick search on webofscience will also provide you with many other references on both meshfree and pu-based methods.

Maybe the XFEM is superior to meshfree methods for treatment of discontinuities, but the strategy of enrichment is problem-dependent. Also, when the elements are cut through by the discontinuities, the subdivision of the element, in general, is needed and should be treated properly. I prefer some implicit treatment strategies of moving discontinuities, such as level set method you mentioned in another post, and also the phase field method. These methods avoid the explicit tracking of the discontinuities and the procedures are general. Nevertheless, an additonal cost of solving an level set equation or phase field equation is needed. So the choice of various method is a individual taste.

I just want to comment here that in fact the X-FEM has been used in conjuction with level sets to model many physical features (cracks, free surfaces, phase interfaces, etc).

With the level set method, in particular, even in the finite difference community you will see special treatment of the numerical scheme near interfaces. I believe it is generally accepted that methods which smear the interface across several grid cells are simply not as accurate.

I am a PhD student in electrical engineering I am preparing a thesis on the study of nonlinear phenomena in electrical devices by meshfree methods.
I wish to ask where to find literature in the case of nonlinearities with fixed point method and meshfree Galerkin method if possible.

The discussion of modelling fracture by the phase field method is interesting to me. Can you direct me to most influencial article on the subject. Also do you know of any review article that compared the explicit discontinuity treatment with the implicit treatment for a large class of fracture problems (brittle, quasi-brittle, ductile, etc)

There are several papers about use of phase field to simulate dynamic crack propagation, particularly in brittle materials. Alain Karma have done very excellent research work on this topic. One of their paper is entitled "phase-field model of Mode 3 dynamic fracture". You can find the paper via arXiv: cond-mat/0105034, 2001.

Generally PIM is not conforming and produces discontinuity in the approximation as domain moves. However there is a way to restor conformability. You may see this reference Liu et al., 2005, "A linearly conforming point interpolation method (LC-PIM) for 2D mechanics problems", International Journal of Computational Methods, 2, No. 4,

I can summarize the basis ideas of the method. I assume that you already know the EFG. Using the local partition of unity concept, we incorporate discontinuous function (Heaviside function H(x)) into the EFG approximation to model the discontinuity due to the crack. So, nodes with domain of influence cut by the crack are enriched by the H(x). It means:

For linear elastic fracture mechanics, since you know the singular field at the crack tip, you can also add them into the approximation. So, nodes whose support contain the crack tip are enriched by the asymptotic functions describing the singular field around the tip.

For numerical integration, the background integration cells are used. To get better accuracy, cells cut by crack are divided into triangles (2D) as done in XFEM.

They called the method, XEFG (eXtended EFG). The method can be applied to nonlinear materials, large deformation and cohesive cracks. The problem of crack iniation based on loss of hyperbolicity can be handles with ease.

Hi, i'm new in the meshfree method. I'm doing my master dissertation in an Auto Adaptive Interface Treatment for the EFGM in eletromagnectic problems. I believe that it can be used for cracks too, just adjust the method. I'm brazilian, and presented in poster format in MOMAG 2010. Each interface has a precision, and for each interface contact it has test points that the mean relative errors between the test points compared to the interfaces precision decide wich interface has to be discreted again. Well, sorry for my english, i didn't practice for a long time haha. I believe that i'll finish my work between July and October. If it could help you, tell me. Thanks.

Hello,
I'm working on modelling of induction heating in electromagnetics with meshfree methods (EFG method).
I want to have explanations on the techniques used in the modelling of
different regions in presence (air, ferromagnetic material,...) in the
global domain of study.
thank you

Hello,
I'm working on modelling of induction heating in electromagnetics with meshfree methods (EFG method).
I want to have explanations on the techniques used in the modelling of different regions in presence (air, ferromagnetic material,...) in the global domain of study.
thank you